Irreducible ring

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

The property of being an ideal for which the quotient ring has this property is: irreducible ideal

Definition

Symbol-free definition

A commutative unital ring is termed irreducible if there do no exist a pair of nonzero ideals whose intersection is the trivial ideal. Equivalently, a ring is irreducible if the zero ideal is an irreducible ideal.

Definition with symbols

A commutative unital ring $R$ is termed irreducible if whenever $I\cap J=0$ for ideals $I$ and $J$ of $R$, then either $I=0$ or $J=0$.

Relation with other properties

Stronger properties

• Field
• Integral domain